1. ## Agoh's Conjecture

Hi everyone

I encountered Agoh's Conjecture:

$

$

When n is prime.

Bn is the n th Bernoulli numer.

How can we extract modulo from fractions?
I red somewhere about minimal residue but i did not understand exacty what it is.
Can you give me some examples (modulo of fractions)?

2. Originally Posted by gdmath
Hi everyone

I encountered Agoh's Conjecture:
$
nB_n\equiv-1 \mod n
$
Is this a question?

(Presumably $B_n$ denotes the $n$ -th Bernoulli number, you should say so)

Note the conjecture is for $n$ prime

CB

3. Sorry i accidently posted the thread without complete it.
The whole statement is:

"
Hi everyone

I encountered Agoh's Conjecture:

When n is prime.

Bn is the n th Bernoulli numer.

How can we extract modulo from fractions?
I red somewhere about minimal residue but i did not understand exacty what it is.
Can you give me some examples (modulo of fractions)?

4. Originally Posted by gdmath
Sorry i accidently posted the thread without complete it.
The whole statement is:

"
Hi everyone

I encountered Agoh's Conjecture:

When n is prime.

Bn is the n th Bernoulli numer.

How can we extract modulo from fractions?
I red somewhere about minimal residue but i did not understand exacty what it is.
Can you give me some examples (modulo of fractions)?

suppose $n = 19$

The 18th Bernoulli Number

$B_{18}$ = $\dfrac{43867}{798}$

from the equation: $nB_{n-1}$

$19 \cdot \dfrac{43867}{798}$ REDUCES TO $\dfrac{43867}{42}$

The modular inverse of 42 with modulus 19 is 5.

43867 x 5 = 219335

$219335 \, \equiv \, 18 \, mod(19) \, \equiv \, -1 \, mod(19)$

The fractions are handled by using the modular inverse.

FYI:
The one millionth Bernoulli number has more than 4.7 million digits in the numerator.
&
The two millionth Bernoulli number has more than 10 million digits in the numerator.

for additional information see: Kellner, B. C. The Equivalence of Giuga’s and Agoh’s Conjectures. 15 Sep 2004.
which can be found here: [math/0409259] The Equivalence of Giuga's and Agoh's Conjectures.